Another way to think of $|x|$ is to imagine an axis (that is, one-dimensional space) and calculate the distance from point $x$ to 0 (zero). In fact $\delta(x,y) = |x-y|$ does nothing else, but measures the distance (an unsigned number) between $x$ and $y$. So you could ask:

Of course, the distance between any $x$ and itself is zero, so $R$ is not reflexive. On the other hand, the distance from $x$ to $y$ is the same as from $y$ to $x$, so we can conclude that it is symmetric. Finally, if we know that the distance between $x$ and $y$ is 2 and between $y$ and $z$ is 2 too, can you guess what is the distance between $x$ and $z$ (or more precisely, is it $2$)?